Pensieve header: The charmed root function.

Unprotect[Power];
((a_)^(1/(p_@n_))):= CharmedRoot[p,n, a]
Protect[Power];
CharmedRoot[p_, n_, a_] := CharmedRoot[p,n] = MinimalBy[
N[(a^(1/n))]Table[(E^(2π I k/n)), {k,n}],
Abs[#-CharmedRoot[p,n]]&
][[1]]

(26^(1/(p1@3)))

b={{1,1}/2};
{LocatorPane[Dynamic[b],Graphics[{Pink,Rectangle[{-1,-1},{1,1}]}]], Dynamic[Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black, Point[r=((b[[1,1]]+I b[[1,2]])^(1/(p@5))); {Re[r], Im[r]}]}, PlotRange->{-1,1}]]}

b1={{1,1}/2};
{LocatorPane[Dynamic[b1],Graphics[{Pink,Rectangle[{-1,-1},{1,1}]}]], Dynamic[Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black, Point[r=((b1[[1,1]]+I b1[[1,2]])^(1/(p1@2))); {Re[r], Im[r]}]}, PlotRange->{-1,1}]]}